L(s) = 1 | + (3.16 − 1.82i)2-s + (2.67 − 4.63i)4-s + (8.69 + 15.0i)5-s + 9.68i·8-s + (55.0 + 31.7i)10-s + (−28.2 − 16.2i)11-s + 42.4i·13-s + (39.0 + 67.7i)16-s + (42.8 − 74.1i)17-s + (−59.9 + 34.5i)19-s + 93.0·20-s − 118.·22-s + (−144. + 83.5i)23-s + (−88.8 + 153. i)25-s + (77.5 + 134. i)26-s + ⋯ |
L(s) = 1 | + (1.11 − 0.645i)2-s + (0.334 − 0.578i)4-s + (0.778 + 1.34i)5-s + 0.428i·8-s + (1.74 + 1.00i)10-s + (−0.773 − 0.446i)11-s + 0.906i·13-s + (0.610 + 1.05i)16-s + (0.610 − 1.05i)17-s + (−0.723 + 0.417i)19-s + 1.04·20-s − 1.15·22-s + (−1.31 + 0.757i)23-s + (−0.710 + 1.23i)25-s + (0.585 + 1.01i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.405216194\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.405216194\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-3.16 + 1.82i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-8.69 - 15.0i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (28.2 + 16.2i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 42.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-42.8 + 74.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (59.9 - 34.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (144. - 83.5i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 254. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-281. - 162. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (172. + 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 182.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 140.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (21.5 + 37.2i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-167. - 96.5i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-55.6 + 96.3i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (78.4 - 45.2i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-524. + 908. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 464. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-158. - 91.3i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-119. - 206. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 375.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-719. - 1.24e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 638. iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.83387944136444833457730570718, −10.42970280698825468586644434901, −9.315620847680988555689285694632, −7.992652213477394197901498423355, −6.85154011140344807921433486188, −5.92869573244583324287742886400, −5.06389588866575566254381333910, −3.71126778764081262588345146200, −2.83271342146512160928331419323, −1.95080360611458942524116236973,
0.73437415076640721835634608423, 2.38711091461479776373782779914, 4.11668031620183788836439497778, 4.82687827924976832473271795592, 5.77112074108379949578211917146, 6.26466517628007432855943842763, 7.84394254086518216042311766496, 8.450753719486292291864264774017, 9.920362600215898660437902685771, 10.17587616884717596597824200257